Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C22×#1334 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #102 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1074 conjugacy classes of subgroups; only the 402 normal subgroups are shown.
This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 402 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2);(3,4);(5,6,7,9,8,10,11,12)(14,18,19,26)(15,21)(16,24)(20,22,25,27)(23,28);(13,14,17,19)(15,22,23,27)(16,25,24,20)(18,21,26,28)128 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#12077, C22×#12065, C22×#11651, … |
| maximal normal subgroup of | C22×#12077, C22×#12065, C22×#11651, … |
| maximal quotient of | C22×#7178 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388256);